Lemma 59.87.6. Let $f : X \to S$ be a flat morphism of schemes such that for every geometric point $\overline{x}$ of $X$ the map
\[ \mathcal{O}_{S, f(\overline{x})}^{sh} \longrightarrow \mathcal{O}_{X, \overline{x}}^{sh} \]
has geometrically connected fibres. Then for every cartesian diagram of schemes
\[ \xymatrix{ X \ar[d]_ f & Y \ar[l]^ h \ar[d]^ e \\ S & T \ar[l]_ g } \]
with $g$ quasi-compact and quasi-separated we have $f^{-1}g_*\mathcal{F} = h_*e^{-1}\mathcal{F}$ for any sheaf $\mathcal{F}$ of sets on $T_{\acute{e}tale}$.
Proof.
It suffices to check equality on stalks, see Theorem 59.29.10. By Theorem 59.53.1 we have
\[ (h_*e^{-1}\mathcal{F})_{\overline{x}} = \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}^{sh}) \times _ X Y, e^{-1}\mathcal{F}) \]
and we have similarly
\[ (f^{-1}g_*^{-1}\mathcal{F})_{\overline{x}} = (g_*^{-1}\mathcal{F})_{f(\overline{x})} = \Gamma (\mathop{\mathrm{Spec}}(\mathcal{O}_{S, f(\overline{x})}^{sh}) \times _ S T, \mathcal{F}) \]
These sets are equal by an application of Lemma 59.39.3 to the morphism
\[ \mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}^{sh}) \times _ X Y \longrightarrow \mathop{\mathrm{Spec}}(\mathcal{O}_{S, f(\overline{x})}^{sh}) \times _ S T \]
which is a base change of $\mathop{\mathrm{Spec}}(\mathcal{O}_{X, \overline{x}}^{sh}) \to \mathop{\mathrm{Spec}}(\mathcal{O}_{S, f(\overline{x})}^{sh})$ because $Y = X \times _ S T$.
$\square$
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